In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v...

Question

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

Accepted Answer

Hey, everyone in this problem we have two vectors A and B A is equal to negative 17, I minus nine J and B is equal to negative A I minus 12 J. And we're asked to determine the dot products A dot B and A dot A, we have four answer choices. K. Option A has that first dot product equal to 28 the second equal to 208. Option B has negative 28 negative 208. Option C has 136 and 81 option D has 244 and 370. Let's just get started. And the first thing we want to calculate is a dot B number call when we have a dot Product, what we wanna do is we wanna multiply the corresponding components of each of our vectors and then add them up for each component. OK? So in this case, we have two components, the I component and the J component. So we're gonna have two terms that we add together. The first term is gonna be the first components. So A one B, one plus that second term A two B two. And when we write this, you can imagine that our vectors are in the following format A is equal to A one I plus A two J and B is equal to B one I plus B two J. OK. So the subscript one corresponds to the I components, the twos correspond to the J components. All right. So working this out and we have the I component of A which is gonna be negative 17 multiplied by the I component of B which is negative A. And we're gonna add the J component of A negative nine multiplied by the J component of B which is negative 12. If we simplify, this gives us 136 plus 108 for A dot product between A and B of 244. Hm All right. So looking at our answer choices, we already expect that the answer is going to be D that's the only option that has the correct dot product for A dot B. But we're gonna go ahead and calculate A dot A as well just to double check to make sure we made no mistakes with the first calculation and make sure that we're getting the correct answer for both. So moving to A dot A, we're gonna do it very similarly. OK? But in this case, both vectors are A. So we just get a one multiplied by a one plus A two multiplied by A two. So the I component of a negative 17. So we have negative 17th multiplied by negative plus the J component of A is negative nine. So we get negative nine multiplied by negative nine. We can go ahead and simplify, we get 289 plus and this is gonna be equal to 370. And so the dot product of A with itself gives us 370. So the correct answer here is indeed option D Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

Key Concepts

Dot Product

Vector Components

Magnitude of a Vector

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